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Diese Arbeit befasst sich mit den reduzierten Basismethoden für parametrisierte quasilineare elliptische und parabolische partielle Differentialgleichungen mit stark monotonem Differentialoperator. Es werden alle Bestandteile der Methode mit reduzierter Basis vorgestellt: Basisgenerierung für reduzierte Approximation, Zertifizierung des Approximationsfehlers durch geeignete a-posteriori Fehlerkontrolle und Offine-Online-Zerlegung. Die Methodik wird ferner auf die magnetostatischen und magnetoquasistatischen Näherungen der Maxwellschen Gleichungen angewendet und ihre Gültigkeit wird durch numerische Beispiele bestätigt.
Counts of SARS-CoV-2-related deaths have been key numbers for justifying severe political, social and economical measures imposed by authorities world-wide. A particular focus thereby was the concomitant excess mortality (EM), i.e. fatalities above the expected all-cause mortality (AM). Recent studies, inter alia by the WHO, estimated the SARS-CoV-2-related EM in Germany between 2020 and 2021 as high as 200 000. In this study, we attempt to scrutinize these numbers by putting them into the context of German AM since the year 2000. We propose two straightforward, age-cohort-dependent models to estimate German AM for the ‘Corona pandemic’ years, as well as the corresponding flu seasons, out of historic data. For Germany, we find overall negative EM of about −18 500 persons for the year 2020, and a minor positive EM of about 7000 for 2021, unveiling that officially reported EM counts are an exaggeration. In 2022, the EM count is about 41 200. Further, based on NAA-test-positive related death counts, we are able to estimate how many Germans have died due to rather than with CoViD-19; an analysis not provided by the appropriate authority, the RKI. Through 2020 and 2021 combined, our due estimate is at no more than 59 500. Varying NAA test strategies heavily obscured SARS-CoV-2-related EM, particularly within the second year of the proclaimed pandemic. We compensated changes in test strategies by assuming that age-cohort-specific NAA-conditional mortality rates during the first pandemic year reflected SARS-CoV-2-characteristic constants.
We consider variational discretization of three different optimal control problems.
The first being a parabolic optimal control problem governed by space-time measure controls. This problem has a nice sparsity structure, which motivates our aim to achieve maximal sparsity on the discrete level. Due to the measures on the right hand side of the partial differential equation, we consider a very weak solution theory for the state equation and need an embedding into the continuous functions for the pairings to make sense. Furthermore, we employ Fenchel duality to formulate the predual problem and give results on solution theory of both the predual and the primal problem. Later on, the duality is also helpful for the derivation of algorithms, since the predual problem can be differentiated twice so that we can apply a semismooth Newton method. We then retrieve the optimal control by duality relations.
For the state discretization we use a Petrov-Galerkin method employing piecewise constant states and piecewise linear and continuous test functions in time. For the space discretization we choose piecewise linear and continuous functions. As a result the controls are composed of Dirac measures in space-time, centered at points on the discrete space-time grid. We prove that the optimal discrete states and controls converge strongly in L^q and weakly-* in Μ, respectively, to their smooth counterparts, where q ϵ (1,min{2,1+2/d}] is the spatial dimension. The variational discrete version of the state equation with the above choice of spaces yields a Crank-Nicolson time stepping scheme with half a Rannacher smoothing step.
Furthermore, we compare our approach to a full discretization of the corresponding control problem, precisely a discontinuous Galerkin method for the state discretization, where the discrete controls are piecewise constant in time and Dirac measures in space. Numerical experiments highlight the sparsity features of our discrete approach and verify the convergence results.
The second problem we analyze is a parabolic optimal control problem governed by bounded initial measure controls. Here, the cost functional consists of a tracking term corresponding to the observation of the state at final time. Instead of a regularization term for the control in the cost functional, we consider a bound on the measure norm of the initial control. As in the first problem we observe a sparsity structure, but here the control resides only in space at initial time, so we focus on the space discretization to achieve maximal sparsity of the control. Again, due to the initial measure in the partial differential equation, we rely on a very weak solution theory of the state equation.
We employ a dG(0) approximation of the state equation, i.e. we choose piecewise linear and continuous functions in space, which are piecewise constant in time for our ansatz and test space. Then, the variational discretization of the problem together with the optimality conditions induce maximal discrete sparsity of the initial control, i.e. Dirac measures in space. We present numerical experiments to illustrate our approach and investigate the sparsity structure
As third problem we choose an elliptic optimal control governed by functions of bounded variation (BV) in one space dimension. The cost functional consists of a tracking term for the state and a BV-seminorm in terms of the derivative of the control. We derive a sparsity structure for the derivative of the BV control. Additionally, we utilize the mixed formulation for the state equation.
A variational discretization approach with piecewise constant discretization of the state and piecewise linear and continuous discretization of the adjoint state yields that the derivative of the control is a sum of Dirac measures. Consequently the control is a piecewise constant function. Under a structural assumption we even get that the number of jumps of the control is finite. We prove error estimates for the variational discretization approach in combination with the mixed formulation of the state equation and confirm our findings in numerical experiments that display the convergence rate.
In summary we confirm the use of variational discretization for optimal control problems with measures that inherit a sparsity. We are able to preserve the sparsity on the discrete level without discretizing the control variable.